AI 中文总结
本文利用时空的(1+3)协变分解，在含物质的广义相对论中定义潮汐场与参考系拖曳场为约束方程，揭示了参考系拖曳场等因素对观测者相对速度、陀螺仪自旋及进动的影响。
AI 中文摘要
已知在真空中，两个无限接近的自由下落类时观测者会因潮汐场（即Weyl张量的“电”部分）产生相对加速度；若他们携带惯性制导陀螺仪，则会因参考系拖曳场（即Weyl张量的“磁”部分）观测到陀螺仪之间的相对进动，而参考系拖曳场不存在牛顿引力对应物。本文针对一类携带惯性陀螺仪且构成同余的类时观测者，利用时空的(1+3)协变分解，在存在物质的情况下将这些场定义为约束方程。与牛顿引力不同，由观测者空间分离形成的闭合轮廓上的相对速度（若同余非超曲面正交，还需闭合该轮廓的时间矢量），因参考系拖曳场与流体动量密度的作用，总和不为零；同时，同一轮廓上陀螺仪自旋的相对取向，通常因潮汐场、流体变量（各向异性压强与能量密度）及构成轮廓的分离矢量相关的相对速度空间叉乘的作用，不会相互抵消。此外，研究表明，即使不存在参考系拖曳场，当相邻观测者的相对速度与加速度非平行时，由于微分托马斯进动，他们仍会观测到陀螺仪之间的相对进动。
英文摘要
It is known that in vacuum, two infinitesimally separated timelike observers under free fall notice a relative acceleration due to the tidal field (or the `electric' part of the Weyl tensor), and if they are carrying inertial guidance gyroscopes, they notice relative precession between these gyroscopes due to the frame-drag field (or the `magnetic' part of the Weyl tensor). The frame-drag field thus has no Newtonian analog. In this paper, for a family of timelike observers carrying inertial gyroscopes and forming a congruence, we define these fields as constraint equations in the presence of matter using (1+3) covariant splitting of spacetime. Unlike Newtonian gravity, the relative velocities of the observers around a closed contour formed by their spatial separations (and a temporal vector that closes this contour if the congruence is not hypersurface orthogonal) do not add up to zero due to the frame-drag field and momentum density of the fluid. Moreover, the relative orientations of the gyroscope spins around the same contour do not generally cancel out due to the tidal field, the fluid variables (anisotropic pressure and energy density), and the spatial cross-product of relative velocities associated with the separation vectors that form the contour. Additionally, it is shown that even in the absence of the frame-drag field, neighboring observers see relative precession between their gyroscopes when their relative velocity and acceleration are non-parallel due to differential Thomas precession.
Comments18 pages, 2 figures; subsection on relative Thomas precession further elaborated, 3 footnotes added, minor typos corrected
